To solve this problem, we need to calculate the future amount of bacteria after 8 days, given that the colony increases by 73% every 2 days.
We will use the formula for exponential growth:
[tex]\[ \text{Future Amount} = I \left(1 + r\right)^{ t } \][/tex]
where:
- [tex]\( I \)[/tex] is the initial amount of bacteria,
- [tex]\( r \)[/tex] is the growth rate in decimal form,
- [tex]\( t \)[/tex] is the time period measured in the number of growth periods.
First, let's identify the values from the problem:
- The initial amount of bacteria [tex]\( I \)[/tex] is 150.
- The growth rate [tex]\( r \)[/tex] is 73%, which can be written as 0.73 in decimal form.
- The total number of days is 8 days.
- The growth period is every 2 days.
Next, we need to determine the number of growth periods within the given time frame.
[tex]\[
\text{Number of growth periods} = \frac{\text{Total number of days}}{\text{Growth period}}
\][/tex]
Substitute the values:
[tex]\[
\text{Number of growth periods} = \frac{8}{2} = 4
\][/tex]
This means the bacteria colony will grow 4 times in 8 days. Now, we can apply these values to our formula:
[tex]\[
\text{Future Amount} = 150 \left(1 + 0.73\right)^{4}
\][/tex]
Calculate the growth factor:
[tex]\[
1 + 0.73 = 1.73
\][/tex]
Raise the growth factor to the power of the number of growth periods:
[tex]\[
1.73^4
\][/tex]
Now, multiply this result by the initial amount of bacteria (150):
[tex]\[
\text{Future Amount} = 150 \times 1.73^4 \approx 150 \times 8.95745 = 1343.6175615
\][/tex]
Therefore, after 8 days, the bacteria colony will have grown to approximately 1343.62 microorganisms.
[tex]\[
\text{Future Amount} \approx 1343.62 \text{ microorganisms}
\][/tex]
